#501079

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1062 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,13,23,9,16,28,29,11,8,10,17,12,18,25,27)(7,19,26,36,24,14,30,34,22,33,21,35,20,15,32,31);(1,2)(3,4)(5,7)(6,14)(8,15)(9,24)(10,26)(11,22)(12,20)(13,21)(16,33)(17,34)(18,19)(23,31)(25,30)(27,35)(28,32)(29,36);(2,4)(5,8,11,6)(7,14,22,15)(9,16,12,18)(10,27,13,29)(17,28,23,25)(19,20,33,24)(21,35,26,36)(30,31,32,34);(5,7)(6,15)(8,14)(9,24)(10,26)(11,22)(12,20)(13,21)(16,19)(17,31)(18,33)(23,34)(25,30)(27,36)(28,32)(29,35)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fh,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408275);